I've been considering this sequence:

$$1,2,3,6,12,24,48,96,192,...$$

I've generated the sequence from the rule

$$V_n=\sum_{0\leq i \lt n} V_i$$ $$V_0=1; V_1=2V_0=V_0+V_0$$

What interests me most, is that this sequence - with its rule requiring the sum of a finite, but unbounded, number of components is remarkably similar to a sequence with a local generation rule requiring doubling of the preceding value. In fact, given a reversed finite leftmost subsequence, an arbitrarily long prefix could suggest that the sequence was the reverse of one generated by a local rule (i.e. $V_n=2V_{n-1}|n\ge1$) and the discrepancy would only become apparent at the penultimate value.

It strikes me that this observation should be relevant to all empirical study... as it demonstrates how two fundamentally different underlying models can generate identical values for an infinite number of tests... and that, unless the single critical comparison (between $V_1$ and $V_2$) is made, an inappropriate model can appear to be supported.

Obviously, there are variants on this theme with different values for $V_0$ and $V_1$ - and each stabilises by $V_4$ to match a local doubling rule... and that $V_0=0$ the result is a constant sequence ($V_i=V_1|i\neq0$)... and that even when I chose $V_1\ge V_0$ I see a similar 'anomaly'.

I'm interested to discover other sequences which, when reversed, can appear to have arisen from a different generator function for an arbitrarily large prefix. For example, are there neat sequences that have two equivalent generator functions only for elements after the fifth or later value?

I've been considering this sequence:

$$1,2,3,6,12,24,48,96,192,...$$

I've generated the sequence from the rule

$$V_n=\sum_{0\leq i \lt n} V_i$$ $$V_0=1; V_1=2V_0=V_0+V_0$$

What interests me most, is that this sequence - with its rule requiring the sum of a finite, but unbounded, number of components is remarkably similar to a sequence with a local generation rule requiring doubling of the preceding value. In fact, given a reversed finite leftmost subsequence, an arbitrarily long prefix could suggest that the sequence was the reverse of one generated by a local rule (i.e. $V_n=2V_{n-1}|n\ge1$) and the discrepancy would only become apparent at the penultimate value.

It strikes me that this observation should be relevant to all empirical study... as it demonstrates how two fundamentally different underlying models can generate identical values for an infinite number of tests... and that, unless the single critical comparison (between $V_1$ and $V_2$) is made, an inappropriate model can appear to be supported.

Obviously, there are variants on this theme with different values for $V_0$ and $V_1$ - and each stabilises by $V_4$ to match a local doubling rule... and that $V_0=0$ the result is a constant sequence ($V_i=V_1|i\neq0$)... and that even when I chose $V_1\ge V_0$ I see a similar 'anomaly'.

I'm interested to discover other sequences which, when reversed, can appear to have arisen from a different recurrence relations for an arbitrarily large prefix. For example, are there neat sequences that have two equivalent recurrence relations only for elements after the fifth or later value?